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@INPROCEEDINGS{DiNapoli:868383,
      author       = {Di Napoli, Edoardo},
      title        = {{F}iltering {S}ubspaces: {H}ow {P}arallelism and {HPC} gave
                      new life to an old eigenvalue solver method},
      reportid     = {FZJ-2019-06912},
      year         = {2019},
      abstract     = {Subspace Iteration (SI) is perhaps one of the earliest
                      iterative algorithmsused as a numerical eigensolver. After
                      an early success, SI methods were abandonedin favor of
                      iterative methods having a smaller footprint in terms of
                      FLOP count.In the last 15 years, subspace methods with
                      polynomial and rational filtering haveseen a resurgence. In
                      this talk I illustrate how the advent of HPC
                      middlewaretogether with advanced parallel computing
                      paradigms are at the base of the revivaland success of
                      modern SI methods.Arguably one of the earliest mentions of
                      SI in the scientific literature is thework by L. Bauer in
                      1957, where SI is applied to the solution of the
                      symmetricalgebraic eigenvalue problem. Several were the
                      attempts to further develop andgeneralize it in the 1960s
                      and 1970s. The first notable effort in this direction isthe
                      fundamental work of Rutishauser in a number of papers
                      spanning from 1969 to1970. Rutishauser builds on Bauer’s
                      Simultaneous Iteration method and introducesfor the first
                      time the concept of filtering through Chebyschev
                      polynomials. Startingfrom the mid 1970s, the development of
                      iterative eigensolvers for the Hermitianeigenvalue problem
                      took on a different direction due to the revival of the
                      Lanczosalgorithm and its variants. With respect to the
                      latter, SI methods generally requirea higher FLOP count to
                      reach convergence, and this made them, at the time, nolonger
                      competitive.Subspace iteration eigensolvers with polynomial
                      filtering saw a resurgence inpopularity starting in the
                      middle of the 2000s with their application to the
                      exteriorspectrum of Hamiltonian matrices emerging in
                      electronic structure theory. Soonafter, a different class of
                      SI methods based on rational filters started emerging andsaw
                      a rapid expansion and application to both Hermitian and
                      non-Hermitian eigen-problems. There are two main reasons for
                      the comeback: 1) the advent of highlyspecialized HPC
                      libraries (e.g. BLAS) which are able to make a distinction
                      betweenslow FLOPs and fast FLOPs in the current hierarchy of
                      caches, and 2) the ability to leverage the hierarchy of
                      nested parallelism that methods based on subspace iter-ation
                      can offer. In this talk I present a brief overview of how
                      these two factors havecontributed to the revival of SI,
                      illustrate recent developments in the field, and givean
                      outlook on the future of these methods and on how their use
                      could positivelyimpact scientific computing applications.},
      month         = {Oct},
      date          = {2019-10-28},
      organization  = {13th Workshop on Parallel Numerics,
                       Dubrovnik (Croatia), 28 Oct 2019 - 30
                       Oct 2019},
      subtyp        = {Plenary/Keynote},
      cin          = {JSC},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {511 - Computational Science and Mathematical Methods
                      (POF3-511) / Simulation and Data Laboratory Quantum
                      Materials (SDLQM) (SDLQM)},
      pid          = {G:(DE-HGF)POF3-511 / G:(DE-Juel1)SDLQM},
      typ          = {PUB:(DE-HGF)6},
      url          = {https://juser.fz-juelich.de/record/868383},
}